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arXiv · 1802.07593

From Hamiltonian to zero curvature formulation for classical integrable boundary conditions

Jean Avan·Vincent Caudrelier·Nicolas Crampe
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Abstract

We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1+1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin's formula for the double-row transfer matrix is used to derive Hamilton's equations of motion for both the Lax matrix {\bf and} the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size $2\times 2$ for previously known Hamiltonians of type $BC_N$ and $D_N$ corresponding to constant and dynamical boundary matrices respectively.

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Jean Avan, Vincent Caudrelier, Nicolas Crampe. 2018-06-01. From Hamiltonian to zero curvature formulation for classical integrable boundary conditions. https://doi.org/10.1088/1751-8121%2Faac976

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